I ♥ Icosahedra

Do you love icosahedra?

I do. On Sunday, I talked with a friend about an icosahedron for over an hour. Icosahedra, along with other polyhedra, are a wonderfully accessible entry point into math–and not just simple math, but deep math that gets you pretty far into geometry and topology, too! Just see my previous post about Matthew Wright’s guest lecture.)

A regular icosahedron is one of the five regular surfaces (“Platonic Solids”). It has twenty sides, all congruent, equilateral triangles. Here are three icosahedra:

icosahedron coloringsHere’s a question which is easy to ask but hard to answer:

How many ways can you color an icosahedron with one of n colors per face?

If you think the answer is n^{20}, that’s a good start–there are n choices of color for 20 faces, so you just multiply, right?–but that’s not correct. Here we’re talking about an unoriented icosahedron that is free to rotate in space. For example, do the three icosahedra above have the same coloring? It’s hard to tell, right?

Solving this problem requires taking the symmetry of the icosahedron into account. In particular, it requires a result known as Burnside’s Lemma.

For the full solution to this problem, I’ll refer you to my article, authored together with friends Matthew Wright and Brian Bargh, which appears in this month’s issue of MAA’s Math Horizons Magazine here (JSTOR access required).

I’m very excited that I’m a published author!

Matthew Wright visits RM

Dr. Matthew Wright paid our students a visit this past Friday and gave them a gentle introduction to topology and the Euler Characteristic. This is a topic given little to no treatment inside the traditional K-12 math curriculum, so our students welcomed the opportunity to learn some ‘college math.’ IMG_20140404_111835821He had our students counting vertices, edges, and faces of various surfaces in order to compute the Euler Characteristic. Students discovered that the Euler Characteristic is a topological invariant.

IMG_20140404_111817820In his talk he also walked the students through a proof that there are only five regular surfaces, using the Euler Characteristic. This is more difficult than the typical proof, but elegant because the proof doesn’t appeal to geometry. That is, the proof doesn’t ever require the assumption that the faces, angles, or edges are congruent. In this sense, it is a topological proof.* Very cool indeed!

Matthew's topology guest lecture at RMHSBio: Matthew Wright went to Messiah College and then went on to received his MS and PhD from University of Pennsylvania, where his thesis was in applied and computational topology. He was a professor at Huntington College for two years but is now at the Institute for Mathematics and its Applications at the University of Minnesota for a postdoctoral research fellowship. His hobbies include photography and juggling. On a personal note, Matthew was my roommate in college, and I had the privilege of being his best man in his wedding, as well!

For more about Dr. Wright, visit his website at http://mrwright.org/.

* This proof also appears in the book Euler’s Gem by Dave Richeson.

Chinese bridge inspired by Möbius band


[Guest post by Dr. Chase]

Is THIS bridge pictured above in the shape of a Möbius band or merely “associated” with a Möbius band as the article suggests?  If it is a Möbius band, where is the half-twist?  Do you think that the bridge is beautiful?  The architects have proposed that such a bridge be built in China.

Can you imagine a Möbius band being used for a road?  There was “A subway named Möbius,” to quote the title of a light-hearted 1950 short story by A. J. Deutsch.  It was published in the wonderful 1958 book Fantastia Mathematica.

The bridge above is only a concept.  Other one-sided surfaces have inspired architectural designs that have actually been built.  Here’s a house made in the shape of a Klein bottle.

A bit of mathematical humor.  One person comments on the Klein bottle that he likes the house’s orientation.  Well, if it were a true Klein bottle, it wouldn’t be orientable at all!

Microsoft Equation Editor math font hack

Thanks to a very nice blog commenter, I now know that there is a nice little work-around if you desperately want an equation in a certain font. I’m not sure it’s what I would always want to do, but if you really, really need a particular font, this will work (for a powerpoint or for a poster or some other one-off application).

There’s been a lot of discussion about getting the new Equation Editor to render in different fonts. I love the new equation editor, but I agree that it’s a pain that Microsoft only has one “math font.” (Cambria Math)

Here’s how it works: The following were all produced by creating an equation in the normal way, then selecting the whole equation and changing its format to “Normal Text” (on the Equation Tab). Once you do this, you can go back to the home tab and change the font of the whole equation. Obviously not every symbol will render correctly if that font doesn’t have certain glyphs, but I was surprised how well these rendered.

And here’s sample output (download docx or pdf here):

sample math fonts

A TOK Lecture on Mathematical Thinking

Students in our International Baccalaureate program here at RM are required to take a core class called Theory of Knowledge (TOK) which is kind of a philosophy class for high school students–or, at least the epistemology piece.

In some schools, this course is taught by math teachers. Here at RM, no math teachers currently teach TOK, which is too bad. So I volunteered to put together a guest lecture on Mathematical Thinking. I’ve tried it out once with a TOK class and I gave the lecture for some of my math teacher colleagues today after school. I plan to give the lecture to more TOK classes this spring.

I thought I’d share it with the MTBoS as well, so here it is. Feel free to read, comment on, or borrow my materials. I think other IB math teachers would especially benefit:

One thing that makes my class unique

Photo from Flickr.com, credit Alan Cleaver, under Creative Commons License.

What’s one thing that makes my class unique?

We play Two Truths and a Lie.

Let me explain. I teach 150+ kids each semester (which means I get new ones in January). I used to think that my job was to teach the material, and the kids didn’t need to like me for that mission to be accomplished. It doesn’t matter what they think of me. That’s not my job, so I reasoned. But thanks to reading awesome books like The Essential 55, The Excellent 11 (both by Ron Clark), and most important, Teaching with Love and Logic (Jim Fay and David Funk), I now know that’s completely and totally false. Here’s the truth: You can’t teach students until they like you.

Getting to know my students has become a major part of what teaching means to me now. The Mr. Chase of eight years ago would never have done a get-to-know you activity at all, since it takes valuable instructional time.

The trouble is, it’s super hard to get to know 150 students in one semester. Even learning their names is a monumental task. The cursory get-to-know-you activity on the first day is cool, and better than nothing, but can you really get to know 150 students in ONE DAY? I still do a little mini, fun first-day activity. But here’s an additional, deeper activity that I’ve come to love.

On the first day of class I hand out index cards. I don’t ask students for their information anymore. I can get their parents names, email addresses, phone numbers, address, and more, through our school’s database, just as you probably can. So asking for that information is a waste of time as far as I’m concerned–it’s just busy work for them. Instead, on their index card, I ask them to write their name and Two Truths and a Lie. They can give it to me after the 45 minute period is over. I tell them they can work on it while I’m going over the syllabus, if they find me boring :-). They can even turn it in the next day if they really want to craft an excellent set of statements that will fool their classmates.

Have you ever played this game? Here’s how it works: You write down three statements about yourself, two of which are true and one of which is false. Then people try to guess which statement is the false statement. Students share things that are interesting and unusual–things their closest friends in the class might not even know.

“I speak four languages”

“I have two dogs and a turtle.”

“My grandmother lives in Portugal.”

“I’ve never broken a bone.”

“I’ve been to five continents.”

“I’m a black-belt in Jujitsu.”

“I don’t like chocolate.”

“My dog’s name is Bubbles.”

When you play this at parties, it takes a while–a minute or two for each person. And of course you want to discuss the results afterward. “What languages do you speak??” “Okay, your dog’s name isn’t Bubbles. But do you have a dog? What kind is it? What is its name?”

So if it takes a while, and you want to take your time, how do you fit it into class time? Well, I have a stack of them at the front of the room and whenever we have extra time, throughout the first month or two of school, we pull a random card (or a few) and meet that student. I say “Today we’re going to meet Robert…everyone say hi Robert!” and everyone says “HI ROBERT!!” (way less corny when it actually happens; don’t worry they love it!). Then we read Robert’s card, and on the second reading everyone is required to raise their hand upon hearing the statement they think is false. Great fun. And afterward we ask Robert some follow-up questions.

It’s a fun activity and lets us genuinely get to know one another and learn very unique things about each other. I give them my own Two Truths and a Lie on the first day of class as an example:

1. I’ve done tricks on a flying trapeze.

2. I lived in Peru for a year.

3. My parents have chickens in their backyard.

(Feel free to make guesses as to which of my statements is a lie.)

This was a unique idea to my class, but some of my other teacher friends have adopted it now, so perhaps it doesn’t qualify anymore :-).

This blog post was in response to the prompt, “What is one thing that happens in your classroom that makes it distinctly yours?” which I was encouraged to answer as I participate in the Exploring the MathTwitterBlogosphere challenge. More challenges to come! (And more blog posts, I’m sure!)

Happy Metric Day, by the way!

The first RM math assembly ever

Mr. Chase & Dr. Tanton

A math pep rally is how my administrator described it. So true!

We had a blast hosting Dr. James Tanton yesterday. (Thanks to the USA Science and Engineering Festival and its sponsors for making it possible!) This was certainly the very first “math assembly” in the history of Richard Montgomery High School!

James is a bold man, facing a crowd of 800+ teenagers with only a pen and paper. But his charismatic style was captivating. The kids loved it and I’ve been hearing only good things from all my students.

DSC00854 James talked about his own love for math and how he became a mathematician. He talked about how he was asking mathematical questions long before he ever actually declared himself a mathematician.

SAMSUNG CAMERA PICTURESDSC00862He taught the whole crowd the national math salute and, right from the start, he had us entertained!

DSC00870When he was a kid, James would lie in bed and look up at the tiles in his bedroom and create little mathematical puzzles for himself. He challenged us to solve his puzzles too, and invited a few students up to try their hand at it.


DSC00882We proved an interesting result with James, and unlike most of my proofs, he got a huge round of applause from hundreds of teenagers :-).

SAMSUNG CAMERA PICTURESJames gave our students a real sense of what it’s like to be a mathematician and do mathematical research–it’s a lot like playing! He had the students’ complete attention throughout the assembly and kept them very interested as he walked them through some fun problems and encouraged audience participation. They clapped and cheered for him. Like I said, math pep rally!

DSC00894 Afterward, James spoke with students who were enthusiastically bombarding him with questions, and he even got two autograph requests! (James = Rock star)

SAMSUNG CAMERA PICTURESAfterward, some of the students and some math teachers had lunch with James. James got peppered with some more questions. Did you know his Erdős number is 3? Pretty awesome!

Thank you, James Tanton, for an awesome assembly!